FIXED POINT THEOREMS IN D-METRIC SPACE THROUGH SEMI-COMPATIBILITY. 1. Introduction. Novi Sad J. Math. Vol. 36, No. 1, 2006, 11-19

Transcription

1 Novi Sad J Math Vol 36, No 1, 2006, FIXED POINT THEOREMS IN D-METRIC SPACE THROUGH SEMI-COMPATIBILITY Bijendra Singh 1, Shobha Jain 2, Shishir Jain 3 Abstract The objective of this paper is to introduce the notion of semi-compatible maps in D-metric spaces and deduce fixed point theorems through semi-compatibility using orbital concept, which improve extend and generalize the results of Ume and Kim [8], Rhoades [7] and Dhage et al [6] All the results of this paper are new AMS Mathematics Subject Classification (1991): 54H25, 47H10 Key words and phrases: D-metric space, D-compatible maps, semi-compatible maps, orbit, unique common fixed point 1 Introduction Generalizing the notion of metric space, Dhage [3] introduced D-metric space and proved the existence of a unique fixed point of a self-map satisfying a contractive condition Rhoades [7] generalized Dhage s contractive condition by increasing the number of factors and proved the existence of a unique fixed point of a self-map in a D-metric space Recently, Ume and Kim [8] have introduced the notion of D-compatible maps in a D-metric space and proved the existence of a unique common fixed point of a pair of D-compatible self maps satisfying the contraction of [7] In [2] Cho, Sharma and Sahu introduced the concept of semi-compatible maps in d-topological spaces They define a pair of self-maps (S, T) to be semicompatible if two conditions (i)sy = T y implies ST y = T Sy (ii) {Sx n } x, {T x n } x implies ST x n T x, as n, hold However, (ii) implies (i), taking x n = y and x = T y = Sy So, in D-metric space, we define the semi-compatibility of the pair (S, T) by condition (ii) only The second section of this paper formulates the definition of a semi-compatible pair of self-maps in a D-metric space and discusses its relationship with a D-compatible pair of self-maps with an example While doing so, we observe that, if T is continuous, then (S, T) is D-compatible implies (S, T) is semicompatible However, the semi-compatibility of the pair of (S, T) does not imply its D-compatibility, even if T is continuous (example 21) Hence it is necessary to discuss the existence of common fixed points of semi-compatible pair of self-maps in fixed point theory 1 S S in Mathematics, Vikram University, Ujjain (MP), India 2 Shri Vaisnav Institute of Management, Gumasta Nagar, Indore, India, shobajain1@yahoocom 3 Shree Vaishnav Institute of Technology and Science, Indore (MP), India, jainshishir11@rediffmailcom

2 12 B Singh, S Jain, S Jain In the light of above observations we establish two fixed point theorems in the third section, which generalize, extend and improve the results of [6], [7] and [8] Moreover, these theorems restrict the domain of x, y and also that of boundedness and completeness considerably Further, corollary 34 of our main result improves and corrects the result of Dhage et al [6] 2 Preliminaries Throughout this paper we use the symbols and basic definitions of Dhage [3] In what follows, (X, D) will denote a D-metric space and N stands for the set of all natural numbers Definition 21 Let X be a non-empty set and D : X X X R + (the set of non-negative real numbers) The pair (X, D) is said to be a D-metric space if, (D-1) D(x, y, z) = 0 if and only if x = y = z; (D-2) D(x, y, z) = D(y, x, z) = D(z, y, x) = ; (D-3) D(x, y, z) D(x, y, a) + D(x, a, z) + D(a, y, z), x, y, z, a X Definition 22 Let (X, D) be a D-metric space and S be a non-empty subset of X We define the diameter of S as δ d (S) = Sup{D(x, y, z) : x, y, z S} Definition 23 ([9]) Let T be a multi-valued map on D-metric space (X, D) Let x 0 X A sequence {x n } in X is said to be an orbit of T at x 0 denoted by O(T, x 0 ) if x n 1 T n 1 (x 0 ), i e x n T x n 1, n N An orbit O(T, x 0 ) is said to be bounded if its diameter is finite It is said to be complete if every Cauchy sequence in it converges to some point of X Definition 24 ([3]) A sequence {x n } in a D-metric space is said to converge to a point x X if for ɛ > 0, there exists a positive integer n 0 such that D(x n, x m, x) < ɛ, n, m > n 0 Definition 25 ([3]) A sequence {x n } is said to be a D-Cauchy sequence in X if for each ɛ > 0, there exists a positive integer n 0 such that D(x n, x n+p, x n+p+t ) < ɛ, n > n 0, p, t N Definition 26 ([8]) A pair (S, T ) of self-maps on a D-metric space (X, D) is said to be D-compatible if for all x, y and z X and for some α (0, ) (21) D(ST x, ST y, T Sz) αd(t x, T y, Sz) Definition 27 A pair (S, T ) of self-mappings of a D-metric space is said to be semi-compatible if lim n ST x n = T x, whenever {x n } is a sequence in X such that lim n T x n = lim n Sx n = x In other words, a pair of selfmaps (S, T) is said to be semi-compatible if lim n D(Sx n, Sx n+p, x) = 0 and lim n D(T x n, T x n+p, x) = 0 imply lim n D(ST x n, ST x n+p, T x) = 0

3 Fixed Point Theorems in D-Metric Space through Semi-Compatibility 13 Proposition 21 Let (S, T ) be a D-compatible pair of self maps on a D-metric space (X, D) and T be continuous Then the pair (S, T ) is semi-compatible Proof Let {Sx n } u, {T x n } u To show ST x n T u As T is continuous, T Sx n T u As (S, T ) is D-compatible, for some α (0, ) D(ST x, ST y, T Sz) αd(t x, T y, Sz), x, y, z X Putting x = x n, y = x n+p and z = x n in above condition, we get D(ST x n, ST x n+p, T Sx n ) αd(t x n, T x n+p, Sx n ), which implies lim n D(ST x n, ST x n+p, T u) = 0 Therefore lim n ST x n = T u Hence (S, T ) is semi-compatible Remark 21 In the following example we observe that, (i) The pair of self-maps (S, T ) is semi-compatible yet it is not D-compatible even though T is continuous (ii) The pair (S, T ) is semi-compatible but (T, S) is not semi-compatible (iii) ST = T S, still (T, S) is not semi-compatible Example 21 Let (X, D) be a D-metric space with X = R +, and let D : X X X R + be defined as D(x, y, z) = Max{ x y, y z, z x }, x, y, z X Define self-maps S and T on X as follows: Sx = 0, if x > 0, and S(0) = 1, T x = x, x R +, and x n = 1 n Then Sx n, T x n 0 as n (i) Now, ST x n = Sx n 0 = T (0) ie ST x n T (0) Also as T = I, for any sequence {x n } such that {Sx n } u and {T x n } u, as n, {ST x n } = {Sx n } u(= T u) i e ST x n T u Therefore (S, T ) is semi-compatible Further as T = I, T is continuous Taking x = 0, y = 0 and z = 1 in (21) we get, D(1, 1, 0) αd(0, 0, 0), α (0, ), which is not true Hence (S, T ) is not D-compatible (ii) Now, Sx n, T x n 0 as n, T Sx n = T (0) 0 S(0) Therefore (T, S) is not semi-compatible By (i), ST x n T (0) Therefore (S, T ) is semicompatible (iii) Also, we note that as T = I, ST = T S Thus (S, T ) is commuting yet (T, S) is not semi-compatible Proposition 22 Let S and T be two self-maps of a D-metric space (X, D) such that S(X) T (X) For x 0 X define sequences {x n } and {y n } in X by Sx n 1 = T x n = y n, n N Then O(T 1 S, x 0 ) = {x 0, x 1, x 2,, x n, }, O(ST 1, Sx 0 ) = {y 1, y 2, y 3,,, y n, }

4 14 B Singh, S Jain, S Jain Proof As Sx 0 = T x 1 implies x 1 T 1 Sx 0 and Sx 1 = T x 2 gives x 2 T 1 Sx 1 = (T 1 S) 2 x 0 Similarly, Sx n 1 = T x n gives x n T 1 Sx n 1 = (T 1 S) n x 0 Again, y 1 = Sx 0, y 2 = Sx 1 S(T 1 Sx 0 ) = (ST 1 )Sx 0, y 3 = Sx 2 S(T 1 ST 1 Sx 0 ) = (ST 1 ) 2 Sx 0 Similarly, y n (ST 1 ) n 1 Sx 0 In [5] Dhage introduces the following family of functions: Let Φ denote the class of all functions φ : R + R + satisfying: φ is continuous; φ is non-decreasing; φ(t) < t, for t > 0; n=1 φn (t) <, t R + Before proving the main results we need the following lemmas : Lemma 21 ([5]) Let {x n } X be bounded with D-bound M satisfying D(x n, x n+1, x m ) φ n (M), m > n + 1, where φ Φ Then {x n } is a D-Cauchy sequence in X Lemma 22 Let S and T be two self-maps of a D-metric space (X, D) such that: (I) S(X) T (X); (II) Some orbit {y n } = O(ST 1, Sx 0 ) is bounded; (III) For all x, y, z { O(T 1 S, x 0 ) and for some φ Φ } D(T x, T y, T z), D(Sx, T x, T z), D(Sy, T y, T z), D(Sx, Sy, Sz) φmax D(Sx, T y, T z), D(Sy, T x, T z) Then {y n } is a D-Cauchy sequence in O(ST 1, Sx 0 ) Proof Let x 0 X As S(X) T (X), we can define sequences {x n } and {y n } in X by Sx n 1 = T x n = y n, n N Then D(y n, y n+1, y n+p ) = D(Sx n 1, Sx n, Sx n+p 1 ), D(yn, y φmax n 1, y n+p 1 ), D(y n 1, y n, y n+p 1 ), D(y n+1, y n, y n+p 1 ), D(y n, y n, y n+p 1 ), D(y n 1, y n+1, y n+p 1 ) ie (22) D(y n, y n+1, y n+p ) φmax Again (23) D(y n 1, y n, y n+p 1 ) φmax { D(yn, y n 1, y n+p 1 ), D(y n+1, y n, y n+p 1 ), D(y n, y n, y n+p 1 ), D(y n 1, y n+1, y n+p 1 ) { D(yn 2, y n 1, y n+p 2 ), D(y n 1, y n, y n+p 2 ), D(y n 1, y n 1, y n+p 2 ), D(y n, y n 2, y n+p 2 ) } }

5 Fixed Point Theorems in D-Metric Space through Semi-Compatibility 15 (24) D(y n+1, y n, y n+p 1 ) φmax D(y n, y n 1, y n+p 2 ), D(y n+1, y n, y n+p 2 ), D(y n, y n 1, y n+p 2 ), D(y n 1, y n+1, y n+p 2 ), D(y n, y n, y n+p 2 ) (25) D(y n, y n, y n+p 1 ) φmax {D(y n 1, y n 1, y n+p 2 ), D(y n, y n 1, y n+p 2 )} (26) D(y n 2, y n, y n+p 2 ), D(y n 1, y n 2, y n+p 2 ), D(y n 1, y n+1, y n+p 1 ) φmax D(y n+1, y n, y n+p 2 ), D(y n 1, y n, y n+p 2 ), D(y n 2, y n+1, y n+p 2 ) Substituting (23)-(26) into (22) we get, D(y n, y n+1, y n+p ) φ 2 Max a,b,c {D(y a, y b, y c )}, for all a, b, c such that n 2 a n, n 1 b n + 1, c = n + p 1 Continuing this process it follows that (27) D(y n, y n+1, y n+p ) φ n Max a,b,c {D(y a, y b, y c )}, for all a, b, c such that 0 a n, 1 b n + 1, c = p Let M be the bound of O(ST 1, Sx 0 ) Then it follows from (27) that D(y n, y n+1, y n+p ) φ n (M) Therefore, by Lemma 21, {y n } is a D-Cauchy sequence in O(ST 1, Sx 0 ) 3 Main results Theorem 31 Let S and T be self-maps of a D-metric space (X, D) such that (311) S(X) T (X); (312) The pair (S, T ) is semi-compatible and T is continuous; (313) For some x 0 X, some orbit {y n } = O(ST 1, Sx 0 ) is bounded and complete; (314) For some φ Φ and for all x, y O(T 1 S, x 0 ) O(ST 1, Sx 0 ) and for all z X D(T x, T y, T z), D(Sx, T x, T z), D(Sy, T y, T z), D(Sx, Sy, Sz) φmax D(Sx, T y, T z), D(Sy, T x, T z) Then S and T have a unique common fixed point in X Proof For x 0 X, construct sequences {x n } and {y n } in X as Sx n 1 = T x n = y n, n N Then by Lemma 22, {y n } is a D-Cauchy sequence in O(ST 1, Sx 0 ), which is complete Therefore, (31) y n (= T x n = Sx n 1 ) u X

6 16 B Singh, S Jain, S Jain As T is continuous and (S, T ) is semi-compatible we get, (32) T 2 x n T u, ST x n T u Step 1: Putting x = T x n, y = T x n and z = u in (314) we get D(T T x n, T T x n, T u), D(ST x n, T T x n, T u) D(ST x n, ST x n, Su) φmax D(ST x n, T T x n, T u), D(ST x n, T T x n, T u), D(ST x n, T T x n, T u) Taking limit as n, using (32) we get, which gives D(T u, T u, Su) = 0, (33) T u = Su Step 2: Putting x = x n, y = x n and z = u in (314) we get, D(Sx n, Sx n, Su) φmax Letting n using (31) and (33) we get, D(T x n, T x n, T u), D(Sx n, T x n, T u), D(Sx n, T x n, T u), D(Sx n, T x n, T u), D(Sx n, T x n, T u) D(u, u, Su) φ{d(u, u, Su)} < D(u, u, Su), if D(u, u, Su) > 0, which is a contradiction Therefore D(u, u, Su) = 0, which gives u = Su Hence u = Su = T u ie u is a common fixed point of S and T Step 3: (Uniqueness) Let w be another common fixed point of S and T, then w = Sw = T w Putting x = x n, y = x n and z = w in (314) we get, D(Sx n, Sx n, Sw) φmax Taking limit as n we get, D(T x n, T x n, T w), D(Sx n, T x n, T w), D(Sx n, T x n, T w), D(Sx n, T x n, T w), D(Sx n, T x n, T w) D(u, u, w) φ{d(u, u, w)} < D(u, u, w), if D(u, u, w) > 0, which is a contradiction Therefore D(u, u, w) = 0, which gives u = w Hence u is the unique common fixed point of S and T Remark 31 By (i) of Remark (21) it follows that there are semi-compatible maps (S, T ) which are not D-compatible even if T is continuous The above theorem investigates the common fixed points of such semi-compatible maps (S, T ) in D-metric spaces

7 Fixed Point Theorems in D-Metric Space through Semi-Compatibility 17 In [8], Ume and Kim have proved the following result using contraction of Rhoades [7] : Corollary 31 ([8]) Let X be a complete D-metric space and S and T be self maps on X satisfying : δ d (O S (T x 0 )) < ; S(X) T (X); The pair (S, T ) is D-compatible and T is continuous; For some q [0, 1) and for all x, y, z X, D(T x, T y, T z), D(Sx, T x, T z), D(Sy, T y, T z), D(Sx, Sy, Sz) qmax D(Sx, T y, T z), D(Sy, T x, T z) Then S and T have a unique common fixed point The following corollary is a generalization of it Corollary 32 Let S and T be self-maps of a D-metric space (X, D) satisfying (311), (313), (314) and (331) The pair (S, T ) is D-compatible and T is continuous Then S and T have a unique common fixed point in X Proof Result follows by using Theorem 31 and proposition 21 Remark 32 The above result of [8] is a particular case of Corollary 32 The following theorem is a counterpart of Theorem 31, in which the continuity of S is assumed instead of that of T This also improves the result of [8] Theorem 32 Let S and T be self-maps of a D-metric space (X, D) satisfying (311), (313) and (351) The pair (S, T ) is semi-compatible and S is continuous (352) For some φ Φ, for all x, y O(T 1 S, x 0 ), z X, D(T x, T y, T z), D(Sx, T x, T z), D(Sy, T y, T z), D(Sx, Sy, Sz) φmax D(Sx, T y, T z), D(Sy, T x, T z) Then the self-maps S and T have a unique common fixed point Proof For x 0 X, construct sequences {x n } and {y n } in X as in proof of theorem 31Therefore (31) holds As S is continuous we get ST x n Su, and as(s, T ) is semi-compatible we get ST x n T u As the limit of a sequence is unique we get Su = T u and the rest of the proof follows from steps 2 and 3 of Theorem 31

8 18 B Singh, S Jain, S Jain Remark 33 The above theorem is an improvement of Theorem 31 and also of the result of [8] It (in view of 351) also underlines the exclusive importance of semi-compatibility in fixed point theory In [7] Rhoades has proved the following: Theorem 1 [7]: Let X be a complete and bounded D-metric space and let S be a self-map of X satisfying D(x, y, z), D(Sx, x, z), D(Sy, y, z), D(Sx, Sy, Sz) qmax D(Sx, y, z), D(x, Sy, z) for all x, y, z X and for 0 q < 1 Then S has a unique fixed point p in X and S is continuous at p The following corollary improves and generalizes it by restricting the domains of boundedness, completeness and that of variables x and y to same orbit only Corollary 33 Let S be a self map of a D-metric spaces (X, D) satisfying (371) For x 0 X, an orbit O(S, x 0 ) is bounded and complete; (372) For some 0 q < 1, for all x, y O(S, x 0 ) and z X, D(x, y, z), D(Sx, x, z), D(Sy, y, z) D(Sx, Sy, Sz) qmax D(Sx, y, z), D(x, Sy, z) Then S has a unique fixed point Proof Result follows from Theorem 31 by taking T = I and φ = q(< 1) then (311) and (312) are trivially satisfied and in this case O(T 1 S, x 0 ) O(ST 1, Sx 0 ) = O(S, x 0 ) In [6] Dhage et al prove the following: Theorem 33 ([6]) Let (X, D) be a D-metric space and S be a self map of X Suppose that there exists x 0 X such that O(S, x 0 ) is D-bounded and S is orbitally complete Suppose also that S satisfies D(Sx, Sy, Sz) λmax{d(x, y, z), D(x, Sx, z)}, x, y, z O(S, x 0 ), for some 0 λ < 1 Then S has a unique fixed point in X The following corollary improves, corrects and generalizes this result Corollary 34 Let X be a D-metric space and S be a self-map on X satisfying (361) and (381) D(Sx, Sy, Sz) λmax{d(x, y, z), D(x, Sx, z)}, x, y O(S, x 0 ), z X Then S has a unique fixed point Proof Result follows from Corollary 33 by taking the maximum of first two factors in place of five factors of (372)

9 Fixed Point Theorems in D-Metric Space through Semi-Compatibility 19 Remark 34 The above corollary improves the result of [6] in which x, y and z are taken in O(S, x 0 ) in the contractive condition whereas in the above corollary the domain of x, y is just the orbit O(S, x 0 ), not its closure Also, the domain of z is the whole space X not O(S, x 0 ), for otherwise the uniqueness of the fixed point does not follow This is the correction required in [6] 4 Acknowledgment Authors express deep sense of gratitude to Dr Lal Bahadur Jain, Retired Principal Govt Arts and Commerce College, Indore (M P) India, for his helpful suggestions and cooperation in this work References [1] Ahmad, B, Ashraf, M, Rhoades, B E, Fixed Point for Expansive Mapping in D-Metric Spaces Indian Journal of Pure Appl Math 32 (2001), [2] Cho, Y J, Sharma, B K, Sahu, R D, Semi-Compatibility and Fixed Point Maths Japonica 42 (1995), [3] Dhage, B C, Generalised Metric Spaces and Mappings with Fixed Points Bull Cal Math Soc 84 (1992), [4] Dhage, B C, A Common Fixed Point principle in D-Metric Spaces Bull Cal Math Soc 91 (1999), [5] Dhage, B C, Some Results on Common Fixed Point -I Indian Journal of Pure Appl Math 30 (1999), [6] Dhage, B C, Pathan, A M, Rhoades, B E, A General Existence Principle for Fixed Point Theorems in D-Metric Spaces Internat Journal Math and Math Sci 23 (2000), [7] Rhoades, B E, A Fixed Point Theorem for Generalized Metric Spaces Internat Journal Math and Math Sci 19 (1996), [8] Ume, J S, Kim, J K, Common Fixed Point Theorems in D-Metric Spaces with Local Boundedness Indian Journal of Pure Appl Math 31 (2000), [9] Veerapandi, T, Chandrasekhara Rao, K, Fixed Point Theorems of Some Multivalued Mappings in a D-Metric Space Bull Cal Math Soc 87 (1995), Received by the editors July 12, 2004